More on Lebesgue Integral and Measure



Our objective in this chapter is to add a few more topics to Lebesgue’s theory of measure and integration introduced in Chap. 10. We begin by revisiting the connection to Riemann’s theory and give a short discussion of improper Riemann integrals. Next, we look at integrals depending on a parameter and give sufficient conditions under which the order of limits and integrals may be interchanged as well as conditions that guarantee the possibility of differentiating under the integral sign.


Lebesgue Integral Improper Riemann Integral Absolute Continuity Shadow Points Lebesgue Theory 
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  1. [Gor94]
    Gordon, R.A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics, vol. 4. American Mathematical Society, Providence (1994)Google Scholar
  2. [Rub63]
    Rubel, L.A.: Differentiability of monotone functions. Collquium Mathematicum 10, 276–279 (1963)MathSciNetGoogle Scholar
  3. [Tao11]
    Tao, T.: An introduction to measure theory.

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.MathematicsTowson UniversityTowsonUSA

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